a small dam is constructed across a stream. a vertical cross section of the stream is y=2x^2. The dam is 4 ft tall. set up an integral that estimates the hydrostatic force on the dam when the water is all the way at the top. Force=Pressure*Area. water weighs 62.5 lb/cubic foot

Question

a small dam is constructed across a stream.  a vertical cross section of the  stream is y=2x^2.  The dam is 4 ft tall.  set up an integral that estimates the hydrostatic force on the dam when the water is all the way at the top.  Force=Pressure*Area.  water weighs 62.5 lb/cubic foot

anonymous · Answer

so far i have $$\int\limits_{0}^{4}62.5  $$ i need help with coming up with the Area part of the integral to be evaluated

anonymous · Answer

so far i have $$\int\limits_{0}^{4}62.5  $$ i need help with coming up with the Area part of the integral to be evaluated

dumbcow · Answer

i think you need to find the area of the horizontal cross section since you're integrating from 0 to 4. 
get x in terms of y -> x = sqrt(y/2)
do we know how wide the stream is?

anonymous · Answer

thats what i thought too but i l looked up the definition of vert. cross section and it said from teh side so it makes sense. 
i think the width of the stream is dx since it changes with the depth of the water, looking at the vertical cross section.

anonymous · Answer

ill take a picture and attach what ive drawn

anonymous · Answer

attachment

dumbcow · Answer

ok the area of the inside of the parabola is sum of 2*x distance * dy
->x = sqrt(y/2)
->area = integral 2sqrt(y/2)dy
so add that to integral above,

anonymous · Answer

i got $$\int\limits\limits_{0}^{4}62.5*2x^{2} dx$$  using dx as the width of the estimating rectangle and $$y=2x^{2}$$ as the height of the estimating rectangle

but you're saying it's
$$\int\limits_{0}^{4}62.5*2\sqrt{y/2}dx $$  ?

dumbcow · Answer

correct except its dy not dx
i get an answer of 471.39

anonymous · Answer

why is it dy?

dumbcow · Answer

because we are integrating from 0 to 4 which is the height or y value
imagine that 2sqrt(y/2) is width of a rectangle and dy is the height

anonymous · Answer

yea im just having trouble getting the equation from the problem.  F=Pressure*Area and I got the pressure because it was given and im not understanding what makes up the Area part of the equation

anonymous · Answer

oh ok. x=width and dy is the height because its changing

dumbcow · Answer

correct

anonymous · Answer

thanks!

dumbcow · Answer

its backwards a little bit.
ok your welcome